With terrific craftsmanship, pure storytelling gusto and that Midas-touch ability to find grounds for optimism everywhere, Steven Spielberg has dramatised a true-life cold war spy-swap drama, starring Tom Hanks and Mark Rylance. Those brought up on John Le Carré might perhaps expect from this moral equivalence, shabby compromise and exhausted futility. But Spielberg, with his gift for uncynicism, uncovers decency and moral courage amidst all the Realpolitik. He works from an excellent screenplay by up-and-coming British dramatist Matt Charman, a script punched up in recognisable places by Joel and Ethan Coen.

Bridge of Spies isn't about the Soviets – Spielberg's true target is Guantanamo

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In 1962, America prepared to recover Gary Powers, the U2 spy-plane pilot captured by the Soviets. The plan was hand over their own incarcerated Russian spy Rudolf Abel, in a classic cold war prisoner exchange at dawn on the Glienecke bridge spanning East and West Berlin – the so-called “Bridge of Spies” – with snipers waiting on both sides ready to take their man out in case of last-second betrayal.

Spielberg’s movie shows that the build-up involved agonisingly tense, deniable negotiations in bad faith, with each side calculating and re-calculating, with every day that went past, how likely it was that their man had cracked under interrogation, given up secrets, and therefore become valueless as an asset. To their opponents’ rage the US was actually insisting on a two-for-one: they also wanted an American student named Frederic Pryor wrongfully imprisoned in East Berlin, a deal which would make them look they’d had the best of the bargain.

Austin Stowell plays Gary Powers, Mark Rylance plays the bespectacled and reserved Russian Abel, and Tom Hanks plays the civilian lawyer James Donovan who brokered the whole arrangement almost singlehandedly, and with pure amateur impulsiveness and stubbornness threw Pryor into the mix at the last moment. Donovan had accepted the poisoned chalice of being Abel’s state-sponsored public defender in the first place and persuaded the authorities that in Abel, America had the currency to buy back Powers.

Hanks gives a very satisfying, watchable and assured performance, with just the right amount of hokum, homely and wile in judiciously balanced proportions. Jimmy Stewart gave us Mr Smith Goes to Washington; Hanks gives us Mr Donovan Goes to Cold War Berlin. Where his Donovan is bluffly ingenuous and straightforward, Rylance’s Russian spy Abel is a quietly voiced enigma, greeting the arresting officers in his chaotic Brooklyn apartment dressed in his underwear, asking meekly if he can put in his false teeth. (Details like these lead me to suspect the Coen brothers’ writing hand, and surely it had to be the Coens who created Abel’s bizarre fake “family members” in East Berlin, the phoney nearest-and-dearest the Communists have to produce as a cover for bringing Donovan to the Iron Curtain to start negotiating.)

Repeatedly, Donovan will ask Abel in his prison cell: “Aren’t you worried?” and Abel will deadpan: “Would it help?” Rylance’s gentle, musical voice is a gift for this elegant, repeated gag. Hanks’s Donovan is a straight-arrow American guy, who has to square the circle of believing it to be his constitutional duty to defend a man who is in fact guilty of spying and attempting to undermine the American state – and its constitution. But Hanks shows how almost by accident, Donovan realises that Abel could be Uncle Sam’s ace in the hole.

Once in East Berlin, Donovan has to negotiate the turf-war minefield of Soviet Russia and communist East Berlin, each of which has its own procedure and diplomatic amour propre. He amusingly has to deal with two deeply prickly Iron Curtain personalities: Pryor’s thin-skinned and mercurial lawyer Vogel (Sebastian Koch) and top DDR apparatchik (Burghart Klaussner), each of whom is given to office-based temper tantrums. Again: I wonder if it was the Coens who created these.

The movie also gives us intriguing visual rhymes, duplications perhaps inspired by the Berlin wall. Spielberg withholds the facts about why Powers failed to commit suicide on capture with a poison-pin in a phoney coin – but instead shows us Abel removing secret information from a phoney coin of his own: somehow a dismal, petty skill. Donovan witnesses a horrible event from the window of his Berlin train, and finally sees something weirdly similar from his commuter train into Manhattan. And the subtle, underplayed, expertly directed “chase” sequence on the subway train at the beginning is terrifically good. Bridge of Spies has a brassy and justified confidence in its own narrative flair.

• Bridge of Spies has premiered at the New York film festival. The film is released in the US on 16 October, Australia on 22 October and UK on 27 November